Phase Noise in DDS
—Invited Article —
Phase noise and amplitude noise in DDS
Claudio E.Calosso 9,Yannick Gruson ?,and Enrico Rubiola ?
web page https://www.sodocs.net/doc/0f4782286.html,
9INRIM,Torino,Italy
?CNRS FEMTO-ST Institute,Besanc ?on,France
Abstract —Their article reports on the measurement of phase
noise and amplitude noise of direct digital synthesizers (DDS),ultimately intended for precision time and frequency applications.
The DDS noise S '(f )tends to scale down as 1/?2
0,until the noise hits the limit due to the output stage.The spurs,however disturbing in general,sink power from the white noise.Voltage noise can be more critical in the digital power supply than in the analog supply.Temperature ?uctuations are an issue at 10 3...1Hz Fourier frequency.Passive stabilization (thermal mass)proves to be useful.Other paramours affect the phase noise,like the clock frequency and power.The amplitude 1/f noise is of the order of 110dB(V 2/V 2)/Hz in some reference (typical)conditions.
Owing to the page and ?le size limitations,only a small part of the available data can be published here.An extended and free version of this article is available on https://www.sodocs.net/doc/0f4782286.html, and on https://www.sodocs.net/doc/0f4782286.html, .
I.I NTRODUCTION
After the original article [1]published more than 40years
ago,the DDS is now a mature piece of technology.The devel-opment has been pushed by the semiconductor technology and by applications.The DDS owes its success to the frequency range (dc to GHz and beyond),to the high resolution (μHz),to the fast frequency switching (sub-μs),to the small size and power,and to the suitability to almost-all-digital single-chip implementation.The newcomer can refer to a technical tutorial [2]and to two books [3],[4].
With the availability of single-chip DDSs in the 1980s,phase noise and spurs were on the stage (see for instance [5],[6],[7],[8]),and an exact and computationally
ef?cient
Figure 1.DDS scheme.solution for spurs was found [9].However,a number of experimental issues of phase noise are not addressed properly in the literature,and amplitude noise is totally absent.We target ?lling this gap.
A.Notation
Following the general notation of frequency metrology (see for example [10],[11]),the clock signal is written as v (t )=V 0[1+?(t )]cos [2??0t +'(t )],where ?(t )is the random fractional amplitude,and '(t )is the random phase.As usual,we describe such noise in term of S ?(f )and S '(f ),i.e.,the single-sided power spectral densities of ?(t )and '(t ).The spectra are approximated with the polynomial law,using S ?=P i h i f i and S '=P i b i f i
.
II.DDS OPERATION
Referring to the block diagram of Fig.1and to the state representation of Fig.2,the DDS is governed by
n k =(n k 1+N )mod D ,
(1)
Figure 2.DDS state variable,and its relationships to the output phase.
The state variable n is converted into sinus and cosine by the look-up table,and further converted into the output analog signals by the two DACs.The look-up table is functionally a read-only memory,though the implementation is quite different.The design is generally determined by the high speed,made possible by the low normalized slew rate of the sinusoid(2?),which makes small the difference between contiguous data.
a)Frequency resolution:Since the control word N can be set with the resolution of one,the frequency resolution is
?res=1
D?s.(3)
For example,a48bit DDS has D=2m=2.8?1014,thus a resolution of3.6μHz at1GHz clock frequency.Such resolution is suf?cient for virtually all practical applications, and also gives the opportunity to implement accurate phase or frequency modulation.Owing to the low cost and power of the digital circuits,there are little reasons to choose a signi?cant smaller value of m.For instance,with m=32 bits the resolution would be of233mHz at1GHz clock.So, most DDSs have m=48bits,while32bits is reserved to extremely-high frequency implementations,or then the power is a critical issue.Other values are seldom encountered.
III.N OISE AND SPURS
A.The Egan model for phase noise
Figure3shows a model for phase noise in frequency synthesis,inspired to the Egan’s article on digital dividers[12]. Assuming that the synthesizer core is noise free,the input phase-time x—i.e.,the time jitter—is transferred from the input to the output as it is.Since the unit of angle scales up as D/N,S'(f)scales down as(N/D)2.Of course,this also applies to the noise of the input stage.Below a given?0,the input noise scaled down hits the phase noise of the output stage,set by the SNR(white)and by the up-conversion of near-dc noise(?icker and temperature?uctuations). However,there is a signi?cant difference between the simple digital divider and the DDS.The divider samples the phase noise at the rising edges of the output,at the rate?0=?s/D (the divider has N=1).In the cases of white noise this increases S'(f)by a factor of D,hence overall S'(f)scales down as1/D instead of1/D2.Conversely,the DDS samples the output at the full clock rate?0.Therefore S'(f)scales down as1/D2.
B.Quantization noise
Assessing the resolution,we notice that the number q of DAC bits is the most severe technical limitation,related to the sampling frequency.For reference,at1GHz maximum clock we?nd14bit converters(Analog device AD9912).
The voltage associated to the least signi?cant bit is V LSB= V FSR/2q,where FSR is the(peak-to-peak)dynamic range. The symbols V LSB and V FSR are consistent with the technical literature.If q is large enough(at least8bits)and the signal uses the dynamic range ef?ciently,the quantization noise has rectangular distribution in±1
2
V LSB.This is granted by the Wiener-Khintchine theorem for stationary ergodic systems, which states that the statistical properties can be calculated interchangeably in the ensemble or in the time domain.The variance is 2=1
12
V2LSB=V2FSR/(12?22q).Assuming that the quantization noise is a true random process with no structure, by virtue of the sampling theorem the noise spectrum is uni-formly distributed from0to the Nyquist frequency B=1
2
?s. The Parseval theorem states that the power calculated in the time domain and in the spectrum is the same.Denoting the white noise level with N,the Parseval theorem gives
N=
V2FSR
6?22q?s
.(4)
The signal power is1
8
V2FSR.Thus,the white phase noise b0= N/P0turns into
b0=
4
3
1
22q?s
rad2/Hz.(5) The results discussed in this paragraph derive from a seminal article by Bennet[13],adapted to phase noise.For example, a14bit coverer at the sampling frequency?s=400MHz yields a white noise of 169dBrad2/Hz.
C.DAC resolution
Given the32–48bit resolution of the phase accumulator, it is obvious that the DAC resolution cannot be that high. The DAC has a number q of bits that is determined by the frequency and by the available technology.Accordingly,the
Figure 4.A simulation shows a reduction in the noise ?oor related to the presence of some spurs.
state variable must be truncated by discarding the appropriate number m p of bits.The minimum number p of bits at the input of the look-up table is determined by the need of preserving the DAC resolution,and estimated using the fact that the slew rate of the sinusoid,normalized to ?s and to the full-scale range V FSR is equal to ?.Hence,the digital resolution at the look-up table input must be p min =q +log 2(?),rounded to p min =q +d log 2(?)e =q +2.The value p =q +5is often encountered.
b)Truncation and spurs:Deriving Eq.(4),we assumed that the quantization noise is white,as it happens in most data acquisition systems.The DDS is more complex because the truncation of the state variable (p For the sake of heuristic reasoning,we assume that the phase error is determined only by p .letting aside the relationship between p and q .For example,a 48bit DDS (m =48)may have p =16.Accordingly,two contiguous values of the analog output are separated by 2m p 1=248 16 1'4.3?109invisible values of n . The rotating vector (angle ?k of Fig.2)gives a sawtooth-like phase error distributed in ±? 2m p ,and sampled at the 22.5 M H z : b –1 = –114.5 d B Balun and MAV-11 at the DDS output 11.25 M H z : b –1 = –120.5 d B 5.625 M H z : b –1 = –126 d B 2. 81 M H z : b –1 ≈ –130 d B 1.406 M H z : b –1 ≈ –1 33.5 d B b 0 ≈ –155 dB b 0 ≈ –159.5 dB b 0 ≈ –162.5 dB DDS internal stages (DDS) output stage 30 dB/dec thermal effect? power supply Figure 5.AD9854phase noise measured at different output frequencies. INRIM 10 0 M H z , b –1 = –1 09 .3 d B 25 M H z , b –1 = –1 22.5 d B 6.25 M H z , b –1 = –128.5 d B 3.6 25 M H z , b –1 = –124.2 d B Figure 6.AD9912phase noise measured at different output frequencies. phase error has a long period referred to as the grand repetition rate GRR =D gcd(N ,D )1 ?c .D.Nonlinearity The DAC nonlinearity generates harmonics,integer multi-ples of ?0.The harmonics that exceed the Nyquist frequency 1 ?s are brought to the output bandwidth by aliasing. –140 dB@ 180 MHz:(opa AD8002 white) b –1 = –118 dB,scales as 1/ν2 I-Q PM noise. Take away 3 dB for 2 equal outputs (DACs) hits b –1 = –132 dB INRIM 11.25 M H z : b –1 = –129 d B ?22.5 M H z : b –1 = –124 d B 45 M H z : b –1 = –118 d B ≤ 5. 6 M H z : b –1 ≈ –132 d B ? Figure 8.AD9854phase noise measured at different output amplitudes ?icker noise scales down as the output frequency ( 6dB per factor-of-two),until it hits the noise of the output stage.Unfortunately,both ?gures suffer from noise in the supply line,which was ?xed later. The AD9854has two outputs in quadrature,which we exploit to measure the phase noise of the output stage.In fact,the I Q phase noise contains only the noise of the two DACs,including their own clock circuits,and the output buffer.Looking at Fig.7,the ?icker is generally 10dB lower than that of the whole DDS,but it hits the same value at low output frequency.This con?rms our conclusion that the lowest ?icker in Figures 5and 6is that of the output stage. Unfortunately,this experiment does not reveal the DDS white noise.This happens because the measurement was done with an AD8002output buffer instead of a RF ampli?er.This choice was made for different purposes,because the AD8002provides kHz output,at the cost of large white noise. Figure 8shows the AD9854phase noise measured at different output amplitude levels.While the ?icker is constant, voltage?uctuation already discussed.Two thermal constants are clearly identi?ed in the transient,10s and1m,likely due to the DDS chip and to the heat sink.After the transient, the phase shift is due to the induced temperature change.The reason is that the oven heats the whole board and cables,while the self-heating acts only on the DDS chip.The results are in a fair agreement for the two methods.Therefore,the additional sensitivity due to the card and cables is smaller than that of the DDS chip. The AD9912temperature coef?cient(Fig.12)is constant vs.frequency( 2ps/K)above20MHz,and shows a1/?30 slope below.This is the signature of the?uctuation of a digital threshold at high frequency,and likely of some kind of drift in the analog electronics at low frequency. The thermal resistance and capacitance of the heat sink impact on the temperature?uctuations below1Hz,and in turn on phase noise.Figure13shows the phase noise of a AD9854with two different heat sinks,a0.64cm3Al cube, and150cm3?n heat sink originally intended for Peltier cells. The combined effect of the160g mass and the low gradient due to the conductance stabilizes the junction temperature, and reduces the phase?uctuations.Below2Hz,and until approximately1mHz,the large heat sink reduces the phase Figure12.Thermal coef?cient of the AD9912vs.the output frequency. d B r a d 2 / H z f (Hz) ?180 ?170 ?160 ?150 ?140 ?130 ?120 ?110 ?100 ?90 ?80 ?70 ?60 ?50 ?40 ?30 ?20 ?10 0.0001 0.001 0.01 0.1 1 10 ’dds2.dat’ u 1:($2+3) ’bancdds2.dat’ u 1:($2+3) ’dds3.dat’ u 1:($2+3) ’bancdds3.dat’ u 1:($2+3) 0.65 cm3 1.7 g heat sink 150 cm3 / 160 g heat sink –20 d B/ d e c –4 0d B/ d e c b a c k g r o u n d b a c k g r o u n d ?ic k e r DD9854 250 MHz clock 10 MHz output Figure13.Effect of the heat sink on the phase noise spectrum. noise by some10dB/dec or more.Below1mHz,both heat sinks are driven by the room temperature and the stabilization effect vanishes. E.Clock frequency and power. The sampling theorem suggests that the white phase noise is proportional to the reciprocal of the clock frequency,as stated by Eq.(4)and(5).Besides,the integrated-circuit technology sets a bound to the minimum and maximum clock frequency. For example,the AD9912must operated in the200–1000MHz range.A signi?cant degradation in the?icker noise shows up at50MHz and100MHz clock.Conversely,the AD9912 seems to be tolerant to the clock amplitude in a range of at least15dB,with no degradation of phase noise(Fig.14).This exceeds most practical needs. V.A PLITUDE NOISE We measured the AM noise with the scheme of Fig.15, giving special attention to?icker.The method resorts to two previous articles[14],[15].The former describes the problems speci?c to the measurement of AM noise,and the latter reveals secrets and subtleties of the cross-spectrum method.In short, since for small?uctuations the fractional amplitude noise is equal to half the power?uctuation,the Schottky-diode power detector proves to be a suitable transducer.The cross-spectrum method is necessary because,unlike for PM noise,we cannot –163 dB @ –8 dBm ?icker b –1 ≈ –122 dB from supply voltage another ?icker-like process? 1/f4 from thermal ?uctuation? –167 dB @ +7 dBm C l o c k a m p l i t u d e INRIM an AD9852.The AM noise of this synthesizer was measured at European Gravitational Observatory.The AM noise is of the order of 110dBrad 2/Hz .Yet,we could not identify a frequency range where the ?icker noise is lower,as it happens with the AD9854.However,general experience suggests that the AM noise of ampli?ers is signi?cantly lower.Accordingly,the observed AM ?icker is ascribed to the DDS.Acknowledgments We thank Jean-Michel Friedt (Sens′e or,France)and Marco Siccardi (SKK,Italy)for fruitful discussions on digital sys-tems;Federico Paoletti (INFN,Italy)shared his results on AM noise.Symmetricom Inc.borrowed us a TSC5125A for a long time.E.R.is indebted to Giovanni Costanzo and Filippo Levi for inviting at the INRIM in 2011;and to Vincent Giordano for discussions and support over 15years. Ultras.Ferroelec.and Freq.Contr.,vol.37,pp.307–315,July 1990.[13]W.R.Bennett,“Spectra of quantized signals,”Bell Syst.Techn.J., vol.27,July 1948. [14] E.Rubiola,“The measurement of AM noise of oscillators.” https://www.sodocs.net/doc/0f4782286.html,,document arXiv:physics/0512082,Dec.2005. [15] E.Rubiola and F.Vernotte,“The cross-spectrum experimental method.” arXiv:1004.5539[physics.ins-det],Apr.2010.